Game ideals. (English) Zbl 1173.03036

For \(\kappa\) a regular uncountable cardinal and \(\lambda\) a cardinal, \(\lambda>\kappa\), let \(A\subseteq P_{\kappa}(\lambda)\). The author considers the following two-player (I and II) game \(H_{\kappa, \lambda}(A)\): there are \(\omega\) moves and I goes first; I and II alternately choose members of \(P_{\kappa}(\lambda)\). The result is a sequence \(\langle a_n : n<\omega\rangle\). Player I wins if and only if \(\bigcup_{n<\omega} a_n\notin A\). The set of all \(A\) for which player I has a winning strategy in \(H_{\kappa, \lambda}(A)\) is a normal ideal on \(P_{\kappa}(\lambda)\), denoted by \(NG_{\kappa,\lambda}\). The paper under review studies the properties of \(NG_{\kappa,\lambda}\), using such tools as club-guessing, scales, Namba trees, and mutually stationary sets.
The author shows that \(P_{\kappa}(\lambda)\) is the disjoint union of \(M(\kappa,\lambda)\) sets not in \(NG_{\kappa,\lambda}\), where \(M(\kappa,\lambda)\) is \(\lambda^{\aleph_0}\) if \(\text{cf}(\lambda)=\omega\) or \(\text{cf}(\lambda)\geq\kappa\), and \(\lambda^{+}\cdot \lambda^{\aleph_0}\) otherwise. He also observes that if there is no inner model with \(\aleph_2\) measurable cardinals, then \(M(\kappa,\lambda)\) is the least size of any member of the dual filter \(NG_{\kappa,\lambda}^{*}\), and so \(P_{\kappa}(\lambda)\) cannot be decomposed into more than \(M(\kappa,\lambda)\) members of \(NG_{\kappa,\lambda}^{+}\). In the paper’s final section the author proves that if \(2^{<\kappa}\leq \mu^{\aleph_0}\) for some regular cardinal \(\mu\) with \(\kappa<\mu\leq\lambda\), then \(\diamondsuit_{\kappa,\lambda}[NG_{\kappa,\lambda}]\) holds.


03E05 Other combinatorial set theory
03E02 Partition relations
Full Text: DOI


[1] Baumgartner, J.E., Applications of the proper forcing axiom, (), 913-959
[2] Baumgartner, J.E.; Taylor, A.D., Saturation properties of ideals in generic extensions I, Transactions of the American mathematical society, 270, 557-574, (1982) · Zbl 0485.03022
[3] Cummings, J., A model in which GCH holds at successors but fails at limits, Transactions of the American mathematical society, 329, 1-39, (1992) · Zbl 0758.03022
[4] Cummings, J.; Foreman, M.; Magidor, M., Squares, scales and stationary reflection, Journal of mathematical logic, 1, 35-98, (2001) · Zbl 0988.03075
[5] Donder, H.D.; Matet, P., Two cardinal versions of diamond, Israel journal of mathematics, 83, 1-43, (1993) · Zbl 0798.03047
[6] Feng, Q., On stationary reflection principles, (), 83-106 · Zbl 0999.03049
[7] Foreman, M.; Magidor, M., Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on \(P_\kappa(\lambda)\), Acta Mathematica, 186, 271-300, (2001) · Zbl 1017.03022
[8] Friedman, H., On closed sets of ordinals, Proceedings of the American mathematical society, 43, 190-192, (1974) · Zbl 0299.04003
[9] Gitik, M.; Shelah, S., Less saturated ideals, Proceedings of the American mathematical society, 125, 1523-1530, (1997) · Zbl 0864.03031
[10] Kueker, D.W., Countable approximations and Lőwenheim-Skolem theorems, Annals of mathematical logic, 11, 57-103, (1977) · Zbl 0364.02009
[11] Landver, A., Baire numbers uncountable Cohen sets and perfect-set forcing, Journal of symbolic logic, 57, 1086-1107, (1992) · Zbl 0797.03046
[12] Liu, A., Bounds for covering numbers, Journal of symbolic logic, 71, 1303-1310, (2006) · Zbl 1109.03044
[13] Magidor, M., Representing sets of ordinals as countable unions of sets in the core model, Transactions of the American mathematical society, 317, 91-126, (1990) · Zbl 0714.03045
[14] Matet, P., On diamond sequences, Fundamenta mathematicae, 131, 35-44, (1988) · Zbl 0663.03035
[15] Matet, P., Concerning stationary subsets of \([\lambda]^{< \kappa}\), (), 119-127
[16] Matet, P., Covering for category and combinatorics on \(P_\kappa(\lambda)\), Journal of the mathematical society of Japan, 58, 153-181, (2006) · Zbl 1118.03036
[17] Matet, P., Club-guessing, good points and diamond, Commentationes mathematicae universitates carolinae, 48, 211-216, (2007) · Zbl 1199.03031
[18] P. Matet, Guessing with mutually stationary sets, Canadian Mathematical Bulletin (in press) · Zbl 1167.03030
[19] P. Matet, Large cardinals and covering numbers, Fundamenta Mathematicae (in press) · Zbl 1191.03032
[20] P. Matet, The Magidor function and diamond, Preprint · Zbl 1237.03031
[21] P. Matet, Weak saturation of ideals on \(P_\kappa(\lambda)\), Preprint · Zbl 1237.03030
[22] P. Matet, C. Péan, S. Shelah, Cofinality of normal ideals on \(P_\kappa(\lambda)\) I, Preprint
[23] Matet, P.; Péan, C.; Shelah, S., Cofinality of normal ideals on \(P_\kappa(\lambda)\) II, Israel journal of mathematics, 150, 253-283, (2005) · Zbl 1119.03045
[24] Matsubara, Y.; Shelah, S., Nowhere precipitousness of the non-stationary ideal over \(\mathcal{P}_\kappa \lambda\), Journal of mathematical logic, 2, 81-89, (2002) · Zbl 1022.03028
[25] Menas, T.K., On strong compactness and supercompactness, Annals of mathematical logic, 7, 327-359, (1974) · Zbl 0299.02084
[26] Merimovich, C., Extender-based radin forcing, Transactions of the American mathematical society, 355, 1729-1772, (2003) · Zbl 1024.03049
[27] Miller, A.W., The Baire category theorem and cardinals of countable cofinality, Journal of symbolic logic, 47, 275-288, (1982) · Zbl 0487.03026
[28] Namba, K., Independence proof of \((\omega, \omega_\alpha)\)-distributive law in complete Boolean algebras, Commentarii mathematici universitatis sancti Pauli, 19, 1-12, (1971) · Zbl 0263.02035
[29] Namba, K., \((\omega_1, 2)\)-distributive law and perfect sets in generalized Baire space, Commentarii mathematici universitatis sancti Pauli, 20, 107-126, (1971/72) · Zbl 0265.04005
[30] Shelah, S., Cardinal arithmetic, (1994), Oxford University Press Oxford · Zbl 0848.03025
[31] S. Shelah, Non-structure Theory (in press) · Zbl 1029.03021
[32] Shioya, M., Splitting \(\mathcal{P}_\kappa \lambda\) into maximally many stationary sets, Israel journal of mathematics, 114, 347-357, (1999) · Zbl 0955.03047
[33] Solovay, R.M., Real-valued measurable cardinals, (), 397-428 · Zbl 0222.02078
[34] S. Todorcevic, Personal communication
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.